What is the difference between 'Topocentric Positions' and 'Local Resonance'?

Topocentric position calculates the Planets/Moon/Sun to the location on the earths surface, not the earth center (Geocentric). That's only up to about a 1/2 degree change for the Moon, the other planets doesn't even matter.
Local resonance includes the axes (AC/MC), but can be used with geocentric or topocentric planets.
The meaning of topocentric and local is about the same, however.

Which objects are used to calculate the resonant waves?

Sun, Moon, the Ecliptic, and for the local resonance also the AC/MC (year, month, day).
The Solar is an approximate trigger of resonance, the Moon is the finer trigger when aligned with the Solar, the earth rotation is the exact trigger.
The planets define merely the quality raised by these triggers.

Planetary aspects alone will give no exact relation to events if not triggered by Sun and Moon (and a location on earth). That's why the exactness of aspects is not everything, the aspect will be activated by alignment to the Earth/Moon system.

Why are the Sun, Moon, Earth the triggers and not the other planets?

The Solar system is one huge spiral (vortex) with the Sun located at the center (sink). All the planets are located at harmonic positions of this spiral.
The Earth/Moon system is an embedded spiral within the huge spiral of our Solar system.
Resonance within this interference system happens for our Earth, when the Earth/Moon system aligns with harmonics (aspects) of the Solar system. This means, when Sun, Moon, and Earth align (with integer divisions) to planetary constellations. For example, the spring tides at New Moon and Full Moon are an extreme case with a division (harmonic) of 2.
The plane of resonance is mainly the ecliptic, because this is the plane of the solar system.

Sun and Moon as the geometric factor were known to at least one Astro-Meteorologist, George Mc. Cormack. The surface analysis charts of weather repeat in Solar and Lunar cycles, best fit with declination aligned too. This already allows finding historic weather maps with a good similarity. This technique is used by Ken Ring in New Zealand to draw quite good weather maps for every day of the coming year.

Also, the Chinese year is calculated by applying a very similar principle. Here the new year starts at a New Moon around the same time of the year (alignment of Solar + Moon Phase). Moreover, the Chinese Zodiac with it's 12-year cycle combines quite a lot of precession cycles. These two ingredients make Chinese Astrology so accurate, despite it's easy application.